Axiomatizations of team logics

Abstract

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(∼) of Väänänen's firstorder team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.

A Complete Axiomatisation for Quantifier-Free Separation Logic

“…The main properties of these formulae is that their Boolean combinations capture the full logic SLp˚,´q [Loz04a] and all the core formulae can be expressed in SLp˚,´q. Generally speaking, our axiom system naturally leads to a form of constructive completeness, as advocated in [Dou17,Lüc18]: the axiomatisation provides proof-theoretical means to transform any formula into an equivalent Boolean combination 1 J˚pallocpxq^size " 1q ñ pallocpxq^size " 1q˚J (A7)…”

“…Our methodology leads to a calculus that is divided in three parts: (1) the axiomatisation of Boolean combinations of core formulae, (2) axioms and inference rules to simulate a bottom-up elimination of the separating conjunction, and (3) axioms and inference rules to simulate a bottom-up elimination of the magic wand. Such an approach that consists in first axiomatising a syntactic fragment of the whole logic (in our case, the core formulae), is best described in [Dou17] (see also [Wal00,vB11,WC13,Lüc18,DFM19]). Section 7 compares works from the literature with our contribution, either for separation logics (abstract versions, fragments, etc.)…”

Characterizing Downwards Closed, Strongly First-Order, Relativizable Dependencies

“…Another possible negation operator, of clearer interpretation in Team Semantics, is the contradictory negation M |= X ∼ φ ⇔ M |= X φ; but adding it to First-Order Logic together with even very simple dependencies (e.g., constancy atoms) brings the expressive power of the resulting formalism all the way up to Second Order Logic. The logic FO(∼) obtained by taking First-Order Logic (with Team Semantics) and adding to it the contradictory negation (but no dependencies) is however equivalent to First-Order Logic wrt sentences, as shown in [16], and in [36] an axiomatization for it is found. This operator will not be further discussed in this work.…”

“…constancy atoms) brings the expressive power of the resulting formalism all the way up to Second Order Logic. The logic FO(∼) obtained by taking First Order Logic (with Team Semantics) and adding to it the contradictory negation (but no dependencies) is however equivalent to First Order Logic wrt sentences, as shown in [16], and in [36] an axiomatization for it is found. This operator will not be further discussed in this work.…”

Logics for First-Order Team Properties

“…In related previous work, it was shown in [10,13,28] that first-order logic extended with constancy atoms =(x) and FO extended with classical negation ∼ are both equivalent to FO over sentences, whereas on the level of formulas they are both strictly less expressive than FO, and thus fail to capture all first-order team properties. It was also illustrated in [24] that a certain simple disjunction of dependence atoms already defines an NP-complete team property.…”